Renormalization group flows of the $N$-component Abelian Higgs model

Flows of the couplings of a theory of an $N$-component (complex) scalar field coupled to electrodynamics are investigated using the functional renormalization group formalism in $d$ dimensions in covariant gauges. We find charged fixed points for any number of components in $d=3$, in accordance with the findings of [G. Fejos and T. Hatsuda, Phys. Rev. D 93, 121701 (2016)] for $N=1$. It is argued that the appropriate choice of the regulator matrix is indispensable to obtain such a result. Ward-Takahashi identities are analyzed in the presence of the regulator, and their compatibility with the flow equation is investigated in detail.

Figure 1

Structure of the renormalization group flows in $d=3$. Two charged fixed points appear for every $N$. Arrows point from UV to IR.

Figure 2

Fixed points as a function of $N$ in $d=3$. Arrows point into directions where $N$ is increasing.

Figure 3

Flow diagram of the Ginzburg-Landau parameter ${\kappa }_k$ in $d=3$. Arrows point from UV to IR. Above the line of ${\kappa }_{-}$ the phase transition is of second order.